Integrand size = 8, antiderivative size = 27 \[ \int \tan ^3(c+d x) \, dx=\frac {\log (\cos (c+d x))}{d}+\frac {\tan ^2(c+d x)}{2 d} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 3556} \[ \int \tan ^3(c+d x) \, dx=\frac {\tan ^2(c+d x)}{2 d}+\frac {\log (\cos (c+d x))}{d} \]
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Rule 3554
Rule 3556
Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^2(c+d x)}{2 d}-\int \tan (c+d x) \, dx \\ & = \frac {\log (\cos (c+d x))}{d}+\frac {\tan ^2(c+d x)}{2 d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \tan ^3(c+d x) \, dx=\frac {2 \log (\cos (c+d x))+\tan ^2(c+d x)}{2 d} \]
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Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(-\frac {-\left (\tan ^{2}\left (d x +c \right )\right )+\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(28\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}}{d}\) | \(29\) |
default | \(\frac {\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}}{d}\) | \(29\) |
norman | \(\frac {\tan ^{2}\left (d x +c \right )}{2 d}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(31\) |
risch | \(-i x -\frac {2 i c}{d}+\frac {2 \,{\mathrm e}^{2 i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(56\) |
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \tan ^3(c+d x) \, dx=\frac {\tan \left (d x + c\right )^{2} + \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \]
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Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \tan ^3(c+d x) \, dx=\begin {cases} - \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {\tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \tan ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \tan ^3(c+d x) \, dx=-\frac {\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (25) = 50\).
Time = 0.59 (sec) , antiderivative size = 216, normalized size of antiderivative = 8.00 \[ \int \tan ^3(c+d x) \, dx=\frac {\log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) + 1}{2 \, {\left (d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, d \tan \left (d x\right ) \tan \left (c\right ) + d\right )}} \]
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Time = 2.80 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \tan ^3(c+d x) \, dx=\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,d}-\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2\,d} \]
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